Question 1190652: What is the present value of an ordinary annuity having semi-annual payments of 8, 000 pesos for 12 years with an interest rate of 12% compounded annually?
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the present value of the ordinary annuity:
**1. Determine the effective interest rate per payment period:**
Since payments are semi-annual, but the interest is compounded annually we need to find the effective semi-annual rate. We can't simply divide the annual rate by two, instead we find the equivalent rate such that if we compound the semi-annual rate twice we get the annual rate. The formula to convert is:
(1 + annual rate) = (1 + semi-annual rate)^2
(1 + 0.12) = (1 + semi-annual rate)^2
1.12 = (1 + semi-annual rate)^2
√1.12 = 1 + semi-annual rate
1.0583 = 1 + semi-annual rate
semi-annual rate = 1.0583 - 1
semi-annual rate = 0.0583 or 5.83%
**2. Determine the number of payment periods:**
* The annuity lasts for 12 years.
* Payments are made semi-annually, meaning twice a year.
* Number of payment periods (n) = 12 years * 2 payments/year = 24 periods
**3. Use the present value of an ordinary annuity formula:**
PV = PMT * [1 - (1 + r)^-n] / r
Where:
* PV = Present Value (what we want to find)
* PMT = Payment amount per period (8,000 pesos)
* r = Interest rate per period (0.0583 or 5.83% as a decimal)
* n = Number of periods (24)
**4. Calculate:**
PV = 8000 * [1 - (1 + 0.0583)^-24] / 0.0583
PV = 8000 * [1 - (1.0583)^-24] / 0.0583
PV = 8000 * [1 - 0.2452] / 0.0583
PV = 8000 * 0.7548 / 0.0583
PV = 8000 * 12.9468
PV ≈ 103,574.27 pesos
Therefore, the present value of the ordinary annuity is approximately **103,574.27 pesos**.
Answer by ikleyn(52756)  (Show Source):
You can put this solution on YOUR website! .
What is the present value of an ordinary annuity having semi-annual payments
of 8, 000 pesos for 12 years with an interest rate of 12% compounded annually?
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As I read this problem and its solution in the post by @CPhill, it creates
many questions in my mind.
First question: is this account an accumulative ordinary annuity or a sinking fund,
which pays money out and works as ordinary annuity ?
As I see from solution by @CPhill, they treat the problem as a sinking fund, but it is unclear
from the problem's wording in the post, why they choose to threat the problem this way and not the other way.
Second question: the logic of the solution in post by @CPhill is unclear to me.
He calculates an equivalent semi-annual compounding rate to the given annual
compounding rate, and then applies this semi-annual compounding.
But it CONTRADICTS to the problem, since the bank does not make
semi-annual compounding for this account.
So, the scheme, which @CPhill constructs for his solution, DOES NOT WORK
and does not correspond to the problem.
As the problem is posed, it tries to create something complicated,
but the proposed scheme by @PChill does not work.
Formally, the money that are deposited in the middle of the year, should lie
without compounding and wait to the end of the annual period to be accounted for compounding then.
Overwise, it should be discussed in the bank's contract and, accordingly, presented in the task.
So, in my view, this "problem" is a "three-legs horse" .
Third notice is that the @CPhill's answer is 103,574.27 pesos.
My answer, obtained from MS Excel, using the same formula, is 101,999.57 pesos.
I calculated, using Excel with 15 decimals after the decimal point
and made rounding at the end, without making any intermediate rounding.
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